So since the empire collapsed after WW2, everybody has been casting around for some kind of identity, more and more frantically as more and more bits of the empire have sloughed off. Since Brexit, it's got a bit weird, with all politicians making all statements in front of massive union jacks, and I expect it will get weirder still when Scotland leaves.
Hopefully, the national conversation will arrive at the conclusion that heterogeneity is no bad thing, and we should simply devolve power (cultural as well as economic) to a local level rather than trying to enforce a kind of Disneyfied Boris-Johnson style 'Britishness' on everybody, but I'm not holding my breath.
Of course, the mechanism eventually broke, because of the slaughter of WW1 put the lie to the dream and the rise of continent-sized superpowers put it materially out of reach. Since then, coopting lower classes into upper-class ideals has been unfeasible. A replacement emerged in the heyday of socialist movements (everyone eating the same food, working in the same factories), but then died off, as that way of life also disappeared. What followed is not homogeneous, I agree on that, but I think you’re discounting the strength of what was there before.
Actually, a woman whose family tree is almost entirely composed of German, Dutch, and Scandinavian people. No doubt they hanged quite a few English peasants too, after being literally invited to rule over the country; but, you know, pedantry and all that.
I understand the sentiment, but I would counter: there is nothing more British than turning something utterly non-British into a symbol of britishness (fancy some tea?), so trying to measure purity in this field is not very productive.